Fundamental groups of clique complexes of random graphs

Abstract

We study fundamental groups of clique complexes associated to random Erdős–Rényi graphs Γ. We establish thresholds for a number of properties of fundamental groups of these complexes XΓ. In particular, if p=nα, then we show that 4pt1emgdim(π1(XΓ))=cd(π1(XΓ))=1ifα<−12,gdim(π1(XΓ))=cd(π1(XΓ))=2if−12<α<−1130,gdim(π1(XΓ))=cd(π1(XΓ))=∞if−1130<α<−13, asymptotically almost surely (a.a.s.), where gdim and cd denote the geometric dimension and cohomological dimension correspondingly. It is known that the fundamental group π1(XΓ) is trivial for α>−13. We prove that for −1130<α<−13 the fundamental group π1(XΓ) has 2‐torsion but has no m‐torsion for any given prime m⩾3. We also prove that aspherical subcomplexes of the random clique complex XΓ satisfy the Whitehead conjecture, that is, all their subcomplexes are also aspherical, a.a.s.

Large random simplicial complexes, I

“…Here we used the equation (6). Next we may combine the obtained equality with the inductive hypothesis…”

“…A different model of random simplicial complexes was studied by M. Kahle [13] and by some other authors, see for example [6]. These are the clique complexes of random Erdős-Rényi graphs, i.e.…”

Large random simplicial complexes, II; the fundamental group

“…This is equivalent to Definition 9 under an additional assumption that α 2 = 0. Complexes with µ 1 (S) > 1/3 were studies in §5 of [11].…”

“…The multi-parameter model which we discuss here allows regimes controlled by a combination of probability parameters associated to various dimensions. This model includes the well-known Linial -Meshulam -Wallach model [22], [23] as an important special case; as another important special case it includes the random simplicial complexes arising as clique complexes of random Erdős-Rényi graphs, see [20], [11].…”

Vanishing of cohomology groups of random simplicial complexes

“…Equivalently, a non-empty set U ⊆ [n] forms a simplex in X p (n) if and only if U is a clique in the binomial random graph. Topological properties of X p (n) have been studied in [16,25,26]. Another example is the random neighbourhood complex arising from the binomial random graph by letting each non-empty set of vertices that have a common neighbour form a simplex [24].…”